3.54 \(\int \frac{2+3 x+5 x^2}{(3-x+2 x^2)^3} \, dx\)

Optimal. Leaf size=64 \[ -\frac{131 (1-4 x)}{2116 \left (2 x^2-x+3\right )}-\frac{11 (3 x+5)}{92 \left (2 x^2-x+3\right )^2}-\frac{262 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{529 \sqrt{23}} \]

[Out]

(-11*(5 + 3*x))/(92*(3 - x + 2*x^2)^2) - (131*(1 - 4*x))/(2116*(3 - x + 2*x^2)) - (262*ArcTan[(1 - 4*x)/Sqrt[2
3]])/(529*Sqrt[23])

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Rubi [A]  time = 0.0331539, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {1660, 12, 614, 618, 204} \[ -\frac{131 (1-4 x)}{2116 \left (2 x^2-x+3\right )}-\frac{11 (3 x+5)}{92 \left (2 x^2-x+3\right )^2}-\frac{262 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{529 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2)^3,x]

[Out]

(-11*(5 + 3*x))/(92*(3 - x + 2*x^2)^2) - (131*(1 - 4*x))/(2116*(3 - x + 2*x^2)) - (262*ArcTan[(1 - 4*x)/Sqrt[2
3]])/(529*Sqrt[23])

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{2+3 x+5 x^2}{\left (3-x+2 x^2\right )^3} \, dx &=-\frac{11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}+\frac{1}{46} \int \frac{131}{2 \left (3-x+2 x^2\right )^2} \, dx\\ &=-\frac{11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}+\frac{131}{92} \int \frac{1}{\left (3-x+2 x^2\right )^2} \, dx\\ &=-\frac{11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}-\frac{131 (1-4 x)}{2116 \left (3-x+2 x^2\right )}+\frac{131}{529} \int \frac{1}{3-x+2 x^2} \, dx\\ &=-\frac{11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}-\frac{131 (1-4 x)}{2116 \left (3-x+2 x^2\right )}-\frac{262}{529} \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=-\frac{11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}-\frac{131 (1-4 x)}{2116 \left (3-x+2 x^2\right )}-\frac{262 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{529 \sqrt{23}}\\ \end{align*}

Mathematica [A]  time = 0.0280116, size = 51, normalized size = 0.8 \[ \frac{\frac{46 \left (524 x^3-393 x^2+472 x-829\right )}{\left (-2 x^2+x-3\right )^2}+1048 \sqrt{23} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{48668} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2)^3,x]

[Out]

((46*(-829 + 472*x - 393*x^2 + 524*x^3))/(-3 + x - 2*x^2)^2 + 1048*Sqrt[23]*ArcTan[(-1 + 4*x)/Sqrt[23]])/48668

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Maple [A]  time = 0.046, size = 47, normalized size = 0.7 \begin{align*} 4\,{\frac{1}{ \left ( 2\,{x}^{2}-x+3 \right ) ^{2}} \left ({\frac{131\,{x}^{3}}{1058}}-{\frac{393\,{x}^{2}}{4232}}+{\frac{59\,x}{529}}-{\frac{829}{4232}} \right ) }+{\frac{262\,\sqrt{23}}{12167}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{23}}{23}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2+3*x+2)/(2*x^2-x+3)^3,x)

[Out]

4*(131/1058*x^3-393/4232*x^2+59/529*x-829/4232)/(2*x^2-x+3)^2+262/12167*23^(1/2)*arctan(1/23*(-1+4*x)*23^(1/2)
)

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Maxima [A]  time = 1.45189, size = 76, normalized size = 1.19 \begin{align*} \frac{262}{12167} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{524 \, x^{3} - 393 \, x^{2} + 472 \, x - 829}{1058 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x+2)/(2*x^2-x+3)^3,x, algorithm="maxima")

[Out]

262/12167*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/1058*(524*x^3 - 393*x^2 + 472*x - 829)/(4*x^4 - 4*x^3 +
13*x^2 - 6*x + 9)

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Fricas [A]  time = 1.05006, size = 225, normalized size = 3.52 \begin{align*} \frac{12052 \, x^{3} + 524 \, \sqrt{23}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - 9039 \, x^{2} + 10856 \, x - 19067}{24334 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x+2)/(2*x^2-x+3)^3,x, algorithm="fricas")

[Out]

1/24334*(12052*x^3 + 524*sqrt(23)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*arctan(1/23*sqrt(23)*(4*x - 1)) - 9039*x^
2 + 10856*x - 19067)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)

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Sympy [A]  time = 0.19223, size = 61, normalized size = 0.95 \begin{align*} \frac{524 x^{3} - 393 x^{2} + 472 x - 829}{4232 x^{4} - 4232 x^{3} + 13754 x^{2} - 6348 x + 9522} + \frac{262 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{12167} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**2+3*x+2)/(2*x**2-x+3)**3,x)

[Out]

(524*x**3 - 393*x**2 + 472*x - 829)/(4232*x**4 - 4232*x**3 + 13754*x**2 - 6348*x + 9522) + 262*sqrt(23)*atan(4
*sqrt(23)*x/23 - sqrt(23)/23)/12167

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Giac [A]  time = 1.20159, size = 62, normalized size = 0.97 \begin{align*} \frac{262}{12167} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{524 \, x^{3} - 393 \, x^{2} + 472 \, x - 829}{1058 \,{\left (2 \, x^{2} - x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x+2)/(2*x^2-x+3)^3,x, algorithm="giac")

[Out]

262/12167*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/1058*(524*x^3 - 393*x^2 + 472*x - 829)/(2*x^2 - x + 3)^
2